A time-varying reliability analysis method for ballastless track slabs in cold regions under different thermal insulation measures

Abstract

The present invention relates to the technical field of railway ballastless track slabs, and specifically, to a method for analyzing the time-varying reliability of ballastless track slabs in cold regions under different thermal insulation measures, which comprises the following steps: 1. Statistical analysis is performed on the long-term monitoring data of the temperature field of ballastless track in cold regions to obtain the probability distribution of the temperature field of the track structure under different thermal insulation measures; 2. The Latin hypercube sampling method is used to generate random load samples, and the load effect of the track slab under different thermal insulation measures is analyzed based on the finite element model; 3. In combination with the time-varying resistance model of the track slab, the Monte Carlo key sampling method is used to analyze the time-varying reliability of the track slab's crack resistance and bending resistance under different thermal insulation measures. The present invention can better perform time-varying reliability analysis.

Description

Technical Field

[0001] The invention relates to the technical field of railway ballastless track slabs, and in particular to a time-varying reliability analysis method of ballastless track slabs in cold regions under different thermal insulation measures. Background Art

[0002] The ballastless track structure is mainly composed of poor thermal conductors such as concrete, which has poor temperature transfer performance. When subjected to drastic temperature changes in the external environment, it will cause uneven internal temperature distribution and form a large temperature gradient. Affected by solar radiation and the layered structure of the track, the temperature gradient mainly appears in the vertical direction, causing uneven vertical deformation of the track structure, such as warping and arching of the track plate, and at the same time generating large uneven stress inside the structure. When the tensile strength limit of the concrete is reached, tensile failure will occur, forming cracks and continuing to develop. In addition, under repeated changes in external temperature, the pore water inside the concrete structure will repeatedly freeze and melt, resulting in continuous accumulation of internal damage, and the structure gradually transforms from a dense body to a loose body, which is manifested as a continuous decrease in concrete strength, seriously affecting the durability and reliability of the ballastless track service.

[0003] In the face of the current demand for high-quality development of ballastless track in high-speed railways, its stability and reliability in long-term service in cold regions should be accurately grasped. The design service life of the ballastless track structure is 60 years, which requires the track structure to have the ability to complete the predetermined functions (durability, safety, and applicability) within the design service life. Failure of the track structure during service will seriously affect the safety of railway operations. Considering that the changes in external loads and the strength of the structure itself are random, it is a random process related to the time variable. Therefore, it is necessary to introduce the time variable to analyze the load effect and reliability of the ballastless track under the cold environment, reveal the trend of structural reliability changes within the design life, and on this basis, study the effect of various insulation measures on improving reliability to ensure the long-term stable operation of the ballastless track in the cold environment.

[0004] The temperature field of ballastless track is affected by a variety of meteorological factors such as solar radiation, wind speed, and temperature. Various types of insulation materials can effectively prevent internal and external heat transfer through low thermal conductivity or the ability to reflect solar radiation, and maintain the temperature of the structure stable. Regarding the improvement effect of insulation measures on the stability of the structural temperature field, some scholars believe that insulation measures can significantly maintain the internal temperature of concrete and significantly improve its crack resistance; some scholars have studied the insulation effect of polystyrene and extruded polystyrene (XPS) foam plastics in the railway substructure. The results show that insulation materials can effectively reduce the impact of subgrade frost heave on railway structural deformation; some scholars use ballast layer covering the track surface as a ballastless track insulation measure in cold regions. The results show that ballast layer insulation measures can effectively reduce the structural temperature fluctuations caused by external temperature changes and slow down the cooling rate.

[0005] Many scholars have conducted research on the reliability of track structure bearing capacity under different loads. Some scholars constructed structural functional functions by assuming the mean and coefficient of variation of load and resistance parameters, and analyzed the bending reliability of ballastless track slabs using the second-order moment method; some scholars established functional functions of transverse and longitudinal cracking failure of track slabs under train loads based on the multi-layer composite beam model, and solved the structural reliability under various failure modes based on the Monte Carlo method; some scholars set the initial damage of the interlayer section of the ballastless track, considered the randomness of the damage length, height, rail pad stiffness and elastic modulus, and used the response surface method to analyze the structural reliability. The above studies analyzed the reliability of track structures under different failure modes, but only for the current state without considering the time variable. In fact, the load effect and structural resistance are both time-related random processes. It is necessary to consider the time variable to analyze the time-varying reliability of the structure in order to clearly grasp the service status of the structure. Some scholars considered the attenuation of structural resistance during actual operation, and used the high-order moment method to analyze the time-varying reliability of the track plate under the combined action of train and temperature loads in terms of bending and cracking. Some scholars used the quadratic fourth-order moment method to analyze the time-varying reliability of interlayer damage of slab ballastless track, and further analyzed the sensitivity of initial damage of narrow joints, temperature load, etc. to the time-varying reliability of interlayer damage. Some scholars used the finite element method to establish the transverse bending function of CRTSⅢ type track plate under the coupling of train and temperature loads, and considered the degradation of structural resistance, and analyzed its transverse bending time-varying reliability by direct probability integration method. Some scholars used the first order reliability method (first order reliability) for the ballastless track base plate on the bridge, considering the combined action of train load, temperature gradient and bridge deflection. The fatigue time-varying reliability of the base plate was calculated by converting the time-varying problem into a time-invariant problem using the Form method (FORM for short). Some scholars discretized the time variables to avoid the complex solution of random processes, converted the time-varying problem into a time-invariant problem, and calculated the time-varying reliability of the structure using the Monte Carlo importance sampling method.

[0006] Under the continuous action of loads and the gradual degradation of structural resistance, the reliability of the track structure is at great risk of falling below the limit within the design life. The current methods for improving structural reliability are basically focused on improving structural resistance, keeping the structural resistance at a relatively safe level from the perspective of initial design or later maintenance, and there is a lack of relevant research on improving reliability by reducing the structural load effect. In addition, the current research has the following shortcomings in analyzing the reliability of track structures and improving methods:

[0007] 1) Most of them consider random variables as empirical values ​​and construct random variables by assuming load mean, coefficient of variation and distribution, which is different from the actual situation;

[0008] 2) The current reliability improvement methods mainly focus on improving the structure's own resistance, and there is a lack of research on improving structural reliability by reducing load effects. Summary of the invention

[0009] The present invention provides a method for analyzing the time-varying reliability of ballastless track slabs in cold regions under different thermal insulation measures, which can overcome certain or some defects of the prior art.

[0010] According to the present invention, the time-varying reliability analysis method of cold-region ballastless track slab under different thermal insulation measures comprises the following steps:

[0011] 1. Statistical analysis is performed on the long-term monitoring data of the temperature field of ballastless track in cold regions to obtain the probability distribution of the temperature field of the track structure under different insulation measures;

[0012] Second, the Latin hypercube sampling method was used to generate random load samples, and the load effects of the track slab under different insulation measures were analyzed based on the finite element model;

[0013] 3. Combined with the time-varying resistance model of the track slab, the Monte Carlo key sampling method was used to analyze the time-varying reliability of the track slab's crack resistance and bending resistance under different insulation measures.

[0014] Preferably, in step one, the test casting model is monitored for a long time. The test casting model is a full-scale model of a CRTS III slab track, which includes, from top to bottom, a rail bearing groove, a C60 prefabricated track plate, a C40 self-compacting concrete, and a C40 base plate. Seven measuring points are set at the transverse center of the track structure from the top surface of the track plate to the center of the base plate, which are respectively set as Z1 on the top surface of the track plate, Z2 at 1 / 4 of the distance from the track plate, Z3 in the middle of the track plate, Z4 at the junction of the track plate and the self-compacting concrete, Z5 in the middle of the self-compacting concrete layer, Z6 at the bottom of the self-compacting concrete layer, and Z7 in the middle of the base plate. The temperature distribution law inside the track structure is mastered through the positions of the seven temperature measuring points.

[0015] Preferably, in step one, the temperature difference data of the track slab and the self-compacting concrete layer over multiple days are selected, and a frequency distribution histogram of the maximum positive temperature difference of the track structure, a frequency distribution histogram of the maximum negative temperature difference of the track structure and a corresponding Weibull distribution fitting curve are drawn and analyzed.

[0016] Preferably, in step 2, the process of the Latin hypercube sampling method is as follows: Assuming that the number of random variables is K, P is an N×K matrix, each column is composed of different random integers from 1 to N, and the matrix R is an N×K matrix, and the column elements in R are composed of uniform random numbers from 0 to 1, where N is the number of sample groups selected, then the sampling matrix S is:

[0017]

[0018] The sampling matrix contains random values ​​from 0 to 1, which are uniformly distributed in the stratified region of the cumulative distribution function; ij is the i-th row and j-th column element of matrix S, then:

[0019]

[0020] Where: is the jth variable of the i-th group of samples obtained by Latin hypercube sampling, It is the distribution function that the j-th variable of the sampling matrix S conforms to;

[0021] Through the above steps, stratified sampling can be performed on multiple groups of random variables to generate random load samples.

[0022] Preferably, in step 2, based on the train load and the positive and negative temperature gradient random variable samples, they are input into the finite element model for batch calculation, the maximum value of each load effect of the track plate is extracted under each group of samples, and the frequency distribution histogram is drawn to analyze the track plate load effect.

[0023] As a preferred method, the Monte Carlo key sampling method is:

[0024] Let X = (X1, X2, ···X1) be a structural random variable, whose joint probability density is f X (x) = f X (x1,x2...x n ), the structural function function Z = g(x), then the failure probability P f Written as the mathematical expectation value of the failure domain characteristic function:

[0025]

[0026] Where: R n is an n-dimensional vector space; is an n-dimensional vector of random variable combinations (x1, x2...x n ) T ; is the failure domain indicative function, expressed as:

[0027]

[0028] According to the law of large numbers, the sample mean of the failure domain characteristic function approximately represents the expected value of the failure domain characteristic function, that is, P f for:

[0029]

[0030] At this time, by constructing an important sampling function, the centroid of the sample distribution is controlled near the failure domain, reducing the number of sampling times in the above formula; for the selection of the sampling centroid, the design verification point of the functional function is currently selected, and the important sampling function is constructed based on this point. Let the important sampling density function be h Y (·), then the failure probability P f Translates to:

[0031]

[0032] After N samples are extracted according to the above formula, N sample vectors are obtained. Then we get P f The approximate value is:

[0033]

[0034] As a preferred method, the time-varying reliability analysis method is:

[0035] The performance function Z is:

[0036] Z=g(R,S)=RS

[0037] Where: R is the structural resistance; S is the structural load effect, that is, the deformation and displacement caused by various loads acting on the structure;

[0038] The structural reliability of ballastless track will change with the service time t within the design life, so the structural performance function Z needs to be considered as a time-dependent random process. In this case, the above formula can be rewritten as:

[0039] Z(t)=g(R(t),S(t))=R(t)-S(t)

[0040] Where: t is the time variable; R(t) is the time-varying resistance; S(t) is the time-varying load effect;

[0041] Assume that the operation year is t, consider the time-varying load effect distribution S(t) as the maximum value in [0, t], and the time-varying resistance distribution R(t) as the minimum value in [0, t]. At this time, the above formula is transformed into:

[0042]

[0043] By stabilizing the non-stationary random process, the maximum distribution function of the load effect in the t-year operation is obtained as

[0044] F t (x) = [F(x)] n

[0045] Where F(x) is the daily load effect distribution function; n is the number of effective action days in the operation year t, n = d × t, where d is the effective number of days of load action in a year.

[0046] The present invention carries out temperature field monitoring test of ballastless track in cold regions, obtains the probability distribution of temperature field of track structure under different thermal insulation measures, generates random load samples by random sampling method, analyzes the weakening effect of different thermal insulation measures on track plate load effect based on finite element model, considers the weakening effect of thermal insulation material on temperature gradient, establishes time-varying function of track plate anti-cracking and anti-bending in combination with structural resistance degradation model, analyzes the time-varying reliability of track plate anti-cracking and anti-bending under different thermal insulation measures, in order to provide certain reference for long-term safe operation and maintenance of ballastless track in cold environment. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 This is a flow chart of a time-varying reliability analysis method for ballastless track slabs in cold regions under different thermal insulation measures;

[0048] Figure 2 This is the layout diagram of temperature measurement points in the long-term monitoring test of temperature field of ballastless track in cold regions;

[0049] Figure 3(a) is a schematic diagram of the maximum positive temperature difference fitting of the track structure without thermal insulation measures;

[0050] Figure 3(b) is a schematic diagram showing the maximum positive temperature difference fitting of the track structure under the polyurethane foam board insulation measure;

[0051] Figure 3(c) is a schematic diagram showing the maximum positive temperature difference fitting of the track structure under the rubber-plastic board and foam board insulation measures;

[0052] Figure 4(a) is a frequency distribution histogram of the maximum positive temperature difference of the track structure without insulation measures and a schematic diagram of curve fitting;

[0053] Figure 4(b) is a frequency distribution histogram of the maximum positive temperature difference of the track structure under the polyurethane foam board insulation measures and a schematic diagram of the curve fitting;

[0054] Figure 4(c) is a frequency distribution histogram and curve fitting diagram of the maximum positive temperature difference of the track structure under the rubber-plastic board and foam board insulation measures;

[0055] Figure 5(a) is a schematic diagram of the variation amplitude of sample mean under different sample sizes of the LHS method;

[0056] Figure 5(b) is a schematic diagram of the variation amplitude of sample standard deviation under different sample sizes of the LHS method;

[0057] Figure 6 It is a schematic diagram of the finite element solid model of CRTSⅢ type slab ballastless track;

[0058] Figure 7 Schematic diagram of the change of relative tensile strength of concrete under different freeze-thaw cycles;

[0059] Figure 8 It is a schematic diagram of the degradation curve of concrete tensile strength over time;

[0060] Figure 9(a) is a schematic diagram of the time-varying reliability of the longitudinal crack resistance of the track slab under different insulation conditions;

[0061] Figure 9(b) is a schematic diagram of the time-varying reliability improvement value of the longitudinal crack resistance of the track slab under different insulation conditions;

[0062] Figure 10(a) is a schematic diagram of the time-varying reliability of the transverse crack resistance of the track slab under different insulation conditions;

[0063] Figure 10(b) is a schematic diagram of the time-varying reliability improvement value of the track slab's transverse crack resistance under different insulation conditions;

[0064] Figure 11(a) is a schematic diagram of the time-varying reliability of the longitudinal bending resistance of the track slab under different insulation conditions;

[0065] Figure 11(b) is a schematic diagram of the time-varying reliability improvement value of the longitudinal bending resistance of the track slab under different insulation conditions;

[0066] Figure 12(a) is a schematic diagram of the time-varying reliability of the track slab in transverse bending under different insulation conditions;

[0067] Figure 12(b) is a schematic diagram of the time-varying reliability improvement value of the track slab's transverse bending resistance under different insulation conditions. DETAILED DESCRIPTION

[0068] In order to further understand the content of the present invention, the present invention is described in detail in conjunction with the accompanying drawings and embodiments. It should be understood that the embodiments are only for explaining the present invention and are not intended to limit it.

[0069] Example

[0070] like Figure 1 As shown, this embodiment provides a time-varying reliability analysis method for cold-region ballastless track slabs under different thermal insulation measures, which includes the following steps:

[0071] 1. Statistical analysis is performed on the long-term monitoring data of the temperature field of ballastless track in cold regions to obtain the probability distribution of the temperature field of the track structure under different insulation measures;

[0072] Second, the Latin hypercube sampling method was used to generate random load samples, and the load effects of the track slab under different insulation measures were analyzed based on the finite element model;

[0073] 3. Combined with the time-varying resistance model of the track slab, the Monte Carlo key sampling method was used to analyze the time-varying reliability of the track slab's crack resistance and bending resistance under different insulation measures.

[0074] Statistics of temperature gradient distribution of track structure under different insulation measures

[0075] Temperature parameters are important parameters in ballastless track design. Affected by external meteorological conditions and the heat conduction properties of concrete itself, uneven temperature fields are easily generated inside the track structure, forming temperature gradients in different directions and degrees, causing structural components to warp and deform, and generating extremely uneven temperature stresses under constraints, putting the track structure in an unfavorable state of stress and deformation. To understand the specific state of temperature stress inside the track structure, it is first necessary to understand the distribution of temperature fields on the surface and inside of the structure. This embodiment relies on the long-term monitoring test of the temperature field of ballastless track in cold regions, monitors the long-term temperature field distribution data of the track structure under polyurethane foam board and rubber-plastic board insulation materials, and sets a comparison with the situation without insulation.

[0076] like Figure 2 As shown in the figure, the test casting model is monitored for a long time. The test casting model is a full-scale model of CRTSⅢ slab ballastless track, which includes the bearing groove, C60 prefabricated track plate, C40 self-compacting concrete and C40 base plate from top to bottom. In order to realize the internal temperature monitoring of the track structure, the temperature sensor embedded parts are made of the same grade concrete in advance, the measuring point position is fixed, and the temperature sensor is cast together with the surrounding concrete, which greatly reduces the internal temperature measurement error of the track structure. Seven measuring points are set at the transverse center of the track structure from the top surface of the track plate to the center of the base plate, which are set as the top surface of the track plate Z1, the position of 1 / 4 of the track plate Z2, the middle of the track plate Z3, the junction of the track plate and the self-compacting concrete Z4, the middle of the self-compacting concrete layer Z5, the bottom of the self-compacting concrete layer Z6, and the middle of the base plate Z7. Through the seven temperature measuring point positions, the temperature distribution law inside the track structure is mastered.

[0077] The test monitored the track structure temperature data under different insulation conditions from July 2022 to the present, with a test interval of 0.5h. The temperature difference data of the track slab and the self-compacting concrete layer over multiple days were selected to draw the frequency distribution histogram of the maximum positive temperature difference of the track structure, the frequency distribution histogram of the maximum negative temperature difference of the track structure and the corresponding Weibull distribution fitting curve, and analyze them as follows:

[0078] 1) Positive temperature gradient

[0079] In order to obtain the probability distribution characteristics of positive temperature gradient loads under high temperature weather, this embodiment selects the daily temperature data of the inner track slab and self-compacting concrete layer from July to August 2022 and 2023, counts the maximum positive temperature difference of the daily track structure, and draws a frequency distribution histogram after screening outliers in the data.

[0080] It can be seen that the maximum vertical temperature difference of the track structure without insulation is 21.6℃, and the average is 13.12℃. The maximum temperature difference of the structure under polyurethane insulation measures is 9.96℃, which is a decrease of 53.9%, and the average is 6.21℃, which is a decrease of 52.7%. The maximum temperature difference of the structure under rubber-plastic board insulation measures is 10.40℃, which is a decrease of 51.9%, and the average is 6.35℃, which is a decrease of 51.6%. It shows that insulation measures have a significant effect on preventing heat transfer and maintaining system temperature stability, among which polyurethane materials have better thermal insulation effect than rubber-plastic board materials. In the three cases, the overall data are all skewed distribution. For this kind of skewed distribution, Weibull distribution can generally be used for fitting. Weibull distribution is a continuous probability distribution with good flexibility and diversity, and is suitable for applications in various scenarios and forms.

[0081] The maximum temperature difference data of the track structure are fitted based on the Weibull distribution, as shown in Figure 3(a), Figure 3(b) and Figure 3(c).

[0082] It can be seen that the maximum positive temperature difference data sample of the track structure has a good fit with the probability density curve of the Weibull distribution, indicating that the Weibull distribution has a good adaptability to skewed data. The KS test with a significance level of 0.05 was used to test the fitting effect. The results showed that the original hypothesis was not rejected, indicating that the fitting model can fully reflect the distribution characteristics of the original data sample and can be used to simulate the probability distribution of the maximum positive temperature difference of the track structure under different insulation measures.

[0083] The temperature difference inside the track structure is generally distributed in an exponential manner in the vertical direction, and the formula is as follows:

[0084] T=T0e ay

[0085] Where T is the temperature difference between the temperature at different depths of the track structure and the bottom of the self-compacting concrete layer; T0 is the maximum vertical temperature difference of the track structure, that is, the temperature difference between the track slab surface and the bottom of the self-compacting concrete layer; e is a natural constant; a is the vertical temperature coefficient of the track structure, which is obtained by fitting the temperature distribution data; y is the vertical depth of the track structure, that is, the vertical distance between different positions and the track slab surface.

[0086] Based on the average positive temperature difference at different depths of the track structure in three cases, the vertical temperature coefficient a of the track structure is fitted, and the fitting values ​​are shown in Table 1.

[0087] Table 1 Fitting values ​​of vertical temperature coefficient of track structure under positive temperature gradient

[0088] No insulation Polyurethane foam board Rubber and plastic board a -12.11 -12.50 -11.90 <![CDATA[Goodness-of-fit R 2 > 0.9941 0.9823 0.9571

[0089] It can be seen from Table 1 that the a value is about 12 under the positive temperature gradient, and the exponential function fitting curve used is highly consistent with the average value of the positive temperature difference at different depths of the track structure in the three cases, and there is a great correlation, which shows that the fitting function can fully predict the distribution of the positive temperature difference at different depths of the track structure. The above coefficients are used when considering the positive temperature difference load in the future.

[0090] Negative temperature gradient

[0091] In order to obtain the probability distribution characteristics of negative temperature difference in track structure under low temperature weather, based on the study of the spatiotemporal distribution characteristics of the temperature field of ballastless track, the maximum temperature difference data of the inner track slab and self-compacting concrete layer were selected in December 2022, January 2023, December 2023, and January 2024. The frequency distribution histogram of the maximum negative temperature difference of the track structure and the Weibull distribution fitting curve were drawn as shown in Figures 4(a), 4(b), and 4(c).

[0092] It can be seen that the maximum negative temperature difference of the track structure is -10.1℃, and the average is -4.91℃. In comparison, the minimum value and average value of the maximum temperature difference data of the track structure under the polyurethane insulation measure are -5.58℃ and -2.94℃, respectively, which are reduced by 44.7% and 40.1%, respectively. The minimum value and average value of the maximum temperature difference data of the structure under the rubber-plastic board insulation measure are 8.04 and -4.62℃, respectively, which are reduced by 20.4% and 6%. In a low temperature environment, the insulation material can still effectively prevent heat transfer and keep the temperature of the track structure in a relatively stable state, but the rubber-plastic board is less effective at this time and can provide lower insulation capacity. Similar to the distribution law of the maximum positive temperature difference, the overall distribution is skewed. From the fitting of the distribution curve, it can be seen that the data of the maximum vertical negative temperature difference of the track structure is in good agreement with the fitted Weibull distribution probability density curve, and satisfies the KS test at the 0.05 significance level, indicating that the maximum negative temperature difference samples of the track structure obey the Weibull distribution, and the fitted curve can describe the distribution of the maximum negative temperature difference of the track structure under different insulation measures.

[0093] Similar to the vertical temperature distribution pattern of the track structure under positive temperature difference, the exponential distribution function is used to fit the vertical negative temperature difference distribution of the track structure, and the fitting values ​​are shown in Table 2.

[0094] Table 2 Fitting values ​​of vertical temperature coefficient of track structure under negative temperature gradient

[0095] No insulation Polyurethane foam board Rubber and plastic board a -8.49 -8.14 -7.70 <![CDATA[Goodness of fit R 2 > 0.9591 0.9673 0.9789

[0096] It can be seen from Table 2 that the a value is around -8 under negative temperature gradient, and the fitting effect is excellent. The above value is used as the vertical temperature coefficient in the subsequent negative temperature difference load study.

[0097] Load random variable sample simulation

[0098] In this embodiment, the Latin hypercube sampling (LHS) method is used, which can more effectively cover the parameter space, avoid sample aggregation or gap phenomena that may be caused by simple random sampling, reduce simulation errors, and effectively improve sampling efficiency. The LHS method has been widely used in random simulations in various fields.

[0099] The process of the LHS method is as follows: Assume that the number of random variables is K, P is an N×K matrix, each column is composed of different random integers from 1 to N, and the matrix R is an N×K matrix. The column elements in R are composed of uniform random numbers from 0 to 1, where N is the number of sample groups selected. Then the sampling matrix S is:

[0100]

[0101] The sampling matrix contains random values ​​from 0 to 1, which are uniformly distributed in the stratified region of the cumulative distribution function; ij is the i-th row and j-th column element of matrix S, then:

[0102]

[0103] Where: is the jth variable of the i-th group of samples obtained by Latin hypercube sampling, It is the distribution function that the j-th variable of the sampling matrix S conforms to;

[0104] Through the above steps, multiple groups of random variables can be stratified sampled to generate random load samples. Taking the train load probability distribution as an example, the LHS method is used to calculate the amplitude of the estimated value under different sample sizes to determine the number of sampling times, as shown in Figure 5(a) and Figure 5(b).

[0105] It can be seen that the change amplitude of each estimate under the LHS method random sampling decreases rapidly with the increase of sample size, and then tends to be stable, converges to the overall parameter, and the reduction rate is fast. After 200 sampling times, it has gradually stabilized. At this time, the sample mean change amplitude is 0.08, and the sample standard deviation change amplitude is 0.34, which is 0.1% and 3.6% respectively compared with the overall parameter ratio. At this time, the error is small. In order to ensure sampling accuracy and improve calculation efficiency, the LHS method is used in this embodiment to randomly sample the load probability distribution model, and the sampling number is 500 times.

[0106] Analysis of track slab load effects under train and temperature loads

[0107] In order to analyze the load effect of the track structure under the action of train load and temperature load, this embodiment establishes a finite element solid model of the CRTS III slab track based on the finite element analysis software ABAQUS. The CRTS III slab track is a unit slab track structure. It is mainly composed of rails, fasteners, prefabricated track plates, self-compacting concrete layers, intermediate isolation layers (geotextiles) and base plates. In the model, the rail is CHN60, the fastener is the standard WJ-8 fastener system, the fastener spacing is 630 mm, the rail and the track plate are connected, and the fastener is simulated by the spring unit (Spring) in ABAQUS; the track plate, self-compacting concrete layer, and base plate are all modeled by three-dimensional solid units according to the actual size, and their concrete strength grades are C60, C40, and C40 respectively; in order to eliminate the influence of boundary conditions, a model with a total length of 16.94 m, i.e. the length of three track plates, is established. The track plate seam is 70 mm, and the middle track plate is taken as the research object for analysis. The specific parameters of the model are shown in Table 3.

[0108] Table 3 Parameters of finite element model of CRTSⅢ slab track

[0109]

[0110] Based on the parameters of the track structure components in Table 3, the finite element solid model of CRTSⅢ slab ballastless track is established as follows: Figure 6 shown.

[0111] The ballastless track structure is a seamless line, and the rails and the base bed are continuous structures. In order to simulate the real boundary conditions, the two ends of the rails and the bottom of the base bed are fully constrained, and the end surface of the base bed constrains the longitudinal and lateral displacements. Since the track plate and the self-compacting concrete are well bonded, the relative displacement between the two is not considered, so binding (Tie) is used to simulate their interaction. There is an isolation layer between the self-compacting concrete layer and the base plate, making the track superstructure and the base plate a separated structure. Hard contact is used in the model to simulate the normal contact behavior, and the penalty friction form is set in the tangential direction, and the friction coefficient is set to 0.8.

[0112] This embodiment is based on train load and positive temperature and negative temperature gradient random variable samples, which are input into the finite element model for batch calculation. The maximum value of each load effect of the track plate is extracted under each group of samples, and a frequency distribution histogram is drawn to analyze the track plate load effect.

[0113] Frequency distribution histogram of track structure load effect under train load and positive temperature gradient coupling. It can be seen that when the train is coupled with temperature load, the influence of temperature on the track plate load effect is much greater than that of the train load, and its probability distribution type is more similar to the temperature random load sample, showing a skewed distribution. The skew direction of the track plate stress and bending moment distribution is the same as that of the temperature random load sample. The maximum and mean values ​​of the track plate load effect under the three cases are shown in Table 4.

[0114] Table 4 Statistics of track slab load effects under different insulation measures

[0115]

[0116] As can be seen from Table 4, compared with the case without insulation, the load effect of the track slab under the insulation conditions of polyurethane foam board and rubber-plastic board is greatly reduced. Taking the longitudinal bending moment of the track slab as an example, when there is no insulation measure, the maximum value and the average value are 55.40 and 29.52 kN·m, respectively. In comparison, the values ​​under the polyurethane foam board insulation material are 27.00 and 16.00 kN·m, respectively, which are 51.3% and 45.8% lower; the values ​​under the rubber-plastic board insulation material are 28.80 and 18.09 kN·m, respectively, which are 48.0% and 38.7% lower. In the hot summer weather, the insulation measures adopted have a significant effect on weakening the load effect of the track slab, which can greatly reduce the impact of temperature on the track structure. Among them, the insulation effect of polyurethane foam board is better than that of rubber-plastic board.

[0117] Frequency distribution histogram of track structure load effect under the coupling of train load and negative temperature gradient. It can be seen that the probability distribution of track plate load effect under the coupling of train load and negative temperature gradient is also skewed, and the skew direction of track plate stress and bending moment distribution is the same as the random sample of negative temperature gradient. The statistics of track plate load effect under the coupling of train load and negative temperature gradient are shown in Table 5.

[0118] Table 5 Statistics of track slab load effects under the coupling of train and negative temperature gradient load

[0119]

[0120]

[0121] As can be seen from Table 5, compared with the case without insulation, the load effect of the track slab under the conditions of polyurethane foam board and rubber-plastic board insulation is greatly reduced. Taking the longitudinal tensile stress of the track slab as an example, when there is no insulation measure, the maximum value and the average value are 2.13 and 1.35 MPa, respectively. In comparison, the polyurethane foam board insulation material is 1.36 and 0.93 MPa, respectively, which is a decrease of 36.2% and 31.1%, respectively; the rubber-plastic board insulation material is 1.76 and 1.12 MPa, respectively, which is a decrease of 17.4% and 17.0%. In low temperature weather, the insulation measures still have a good effect and can significantly reduce the tensile stress of the track slab, but at this time the insulation effect of the rubber-plastic board insulation material is already low, and the stress reduction of the track slab is less than 20%.

[0122] Probability distribution fitting of track slab load effect

[0123] In order to explore the time-varying reliability of the track slab against cracking and bending under the most unfavorable conditions, the time-varying reliability of the track slab against cracking is calculated based on the distribution of structural tensile stress under train load and positive temperature gradient, and the time-varying reliability of the track slab against bending is calculated based on the distribution of structural bending moment under train load and negative temperature gradient. The Weibull distribution is used to fit the distribution of transverse and longitudinal tensile stress and bending moment of the track slab under different insulation measures, respectively. The fitting parameters are shown in Tables 6 and 7.

[0124] Table 6 Fitting parameters of maximum tensile stress distribution of track slab under different insulation measures

[0125]

[0126] Table 7 Fitting parameters of maximum bending moment distribution of track slab under different insulation measures

[0127]

[0128] Substituting the above parameters into Weibull distribution can obtain the corresponding load effect distribution function, and combining it with the maximum value distribution function of the load effect in the steady process can obtain the time-varying model of each load effect.

[0129] Time-varying resistance distribution model of track slab structure

[0130] It includes the time-varying resistance distribution model of track slab anti-cracking and the time-varying resistance distribution model of track slab anti-bending.

[0131] Time-varying model of track slab crack resistance

[0132] The track slab is a prefabricated reinforced concrete structure with transverse and longitudinal prestressed steel bars, which form a certain pre-compressive stress in the transverse and longitudinal directions of the track slab, and together with the tensile strength of the concrete itself, form the track slab's anti-crack bearing performance. As the service time increases, the prestressed tendons will continue to relax, and in cold areas, the freeze-thaw cycle will also cause the concrete strength to continue to decrease. Therefore, considering the continuous deterioration of the concrete structure and prestressed tendons, a time-varying model of the track slab's anti-crack capacity is established.

[0133] (1) Time-varying model of concrete strength

[0134] The deterioration of concrete structures in cold regions is affected by many factors, among which freeze-thaw cycles are one of the main reasons for the reduction in strength. In order to explore the strength change law of track slab concrete under freeze-thaw cycles, the research team carried out freeze-thaw cycle tests, with C60 concrete columns of 400×100×100mm as the object. The test applied freeze-thaw cycles 300 times, with 1.5h cooling and 1.5h heating as one cycle, and non-destructive measurement of the dynamic elastic modulus of concrete was performed every 50 freeze-thaw cycles.

[0135] The test measured the dynamic elastic modulus strength of C60 concrete columns when the freeze-thaw cycles were 0, 50, 100, 150, 200, 250, and 300 times. As the freeze-thaw cycles increased, the dynamic elastic modulus of the concrete column gradually decreased. When the freeze-thaw cycles increased from 0 to 300 times, the dynamic elastic modulus decreased from 50.75GPa to 45.67GPa, a decrease of 10%. A regression analysis was conducted on the correlation between the dynamic elastic modulus and compressive strength of concrete with different mix ratios in the rapid freeze-thaw test, and the relationship between the two was obtained as follows:

[0136]

[0137] Where: E d is the dynamic elastic modulus of concrete; f cu is the compressive strength of concrete cube.

[0138] The conversion relationship between the standard value of concrete axial tensile strength and the standard value of cube compressive strength is calculated as follows:

[0139]

[0140] Where: f tk is the standard value of concrete tensile strength; f cu,k is the standard value of concrete cube compressive strength; δ is the coefficient of variation; α c2 is the brittleness reduction factor of concrete.

[0141] The dynamic elastic modulus data is transformed to obtain the relative change of concrete tensile strength. Figure 7 shown.

[0142] In order to evaluate the strength change of concrete structures in actual environments with service time, it is necessary to convert the number of indoor and outdoor freeze-thaw cycles, and combine the annual freeze-thaw number of outdoor environments to determine the relevant rules. The relationship between outdoor and indoor freeze-thaw cycles is generally expressed as follows:

[0143]

[0144] Where: T is the number of outdoor freeze-thaw years; k is the ratio coefficient of outdoor to indoor freeze-thaw; n is the number of indoor freeze-thaw cycles; N a is the number of freeze-thaw cycles per year.

[0145] Figure 8 The degradation curve of concrete tensile strength within 60 years is given, and the exponential degradation form commonly used in engineering is used for fitting. In the 60th year, the relative strength has dropped to 68.26%. The fitting function expression is:

[0146]

[0147] Where: ε is the degradation coefficient, obtained by fitting, ε is a random variable with a mean of 6.37×10 -3 , the coefficient of variation is 0.21; t is time, the unit is year.

[0148] Therefore, the degradation function of concrete tensile strength over time can be expressed as:

[0149]

[0150] Where R c0 represents the standard value of initial tensile strength of concrete, which obeys normal distribution. The standard value of axial tensile strength of C60 concrete given in the specification is f tk It is 2.85MPa, and its coefficient of variation is 0.2.

[0151] (2) Time-varying model of prestress

[0152] The prestress loss of the track slab during long-term service is a complex and important mechanical behavior, which mainly comes from three aspects: concrete shrinkage, creep effect, and relaxation of prestressed tendons. During the use of the structure, due to the existence of the above-mentioned prestress loss, the effective prestress in the concrete will gradually decrease. The long-term prestress loss of prestressed structures often shows a faster rate in the early stage of use. As the service time increases, the rate of prestress loss will gradually slow down. The prediction formula for the prestress loss of ballastless track slabs is:

[0153]

[0154] Where: R p0 is the initial value of precompression stress.

[0155] Time-varying model of track slab bending capacity

[0156] The flexural capacity of the track slab needs to fully consider the combined effects of prestressed tendons, ordinary stress-bearing steel bars and concrete in the compression zone. Based on its force balance equation, the initial flexural bearing capacity of the track slab can be expressed as:

[0157]

[0158] Where: f s 、f' s With f p are the design values ​​of tensile strength of ordinary steel bars in the tension zone, compressive strength of ordinary steel bars in the compression zone, and tensile strength of prestressed steel bars; A s , A' s With A p are the cross-sectional areas of ordinary steel bars and prestressed bars in the tension zone and compression zone respectively; f cis the design value of concrete axial compressive strength; b is the width of the bending section; h0 is the effective height of the section; a' s is the distance from the steel bars in the compression zone to the edge of the section.

[0159] Based on the above formula, the time-varying resistance model is established as follows:

[0160]

[0161] In the formula is the degradation function of the track slab bending strength, which can be expressed as:

[0162]

[0163] Where t is the service time of ballastless track, T0 is the starting time of resistance degradation, in years; k1 and k2 are resistance degradation rate parameters.

[0164] Monte Carlo weighted sampling

[0165] The Monte Carlo importance sampling method increases the number of times the sample points fall into the failure domain by changing the sampling center of gravity, thereby obtaining a higher solution accuracy with a smaller number of sampling times and improving the computational efficiency. Specifically: Let X = (X1, X2, ···X1) be a structural random variable, and its joint probability density is f X (x) = f X (x1,x2...x n ), the structural function function Z = g(x), then the failure probability P f Written as the mathematical expectation value of the failure domain characteristic function:

[0166]

[0167] Where: R n is an n-dimensional vector space; is an n-dimensional vector of random variable combinations (x1, x2...x n ) T ; is the failure domain indicative function, expressed as:

[0168]

[0169] According to the law of large numbers, the sample mean of the failure domain characteristic function approximately represents the expected value of the failure domain characteristic function, that is, P f for:

[0170]

[0171] At this time, by constructing an important sampling function, the centroid of the sample distribution is controlled near the failure domain, reducing the number of sampling times in the above formula; for the selection of the sampling centroid, the design verification point of the functional function is currently selected, and the important sampling function is constructed based on this point. Let the important sampling density function be h Y (·), then the failure probability P f Translates to:

[0172]

[0173] After N samples are extracted according to the above formula, N sample vectors are obtained. Then we get P f The approximate value is:

[0174]

[0175] Time-varying reliability analysis method

[0176] The design life of ballastless track is 60 years. During the design life, the structural reliability will change with the service time t, so its structural function Z also needs to be considered as a time-related random process. Specifically, the function Z is:

[0177] Z=g(R,S)=RS

[0178] Where: R is the structural resistance; S is the structural load effect, that is, the deformation and displacement caused by various loads acting on the structure;

[0179] The structural reliability of ballastless track will change with the service time t within the design life, so the structural performance function Z needs to be considered as a time-dependent random process. In this case, the above formula can be rewritten as:

[0180] Z(t)=g(R(t),S(t))=R(t)-S(t)

[0181] Where: t is the time variable; R(t) is the time-varying resistance; S(t) is the time-varying load effect;

[0182] Assume that the operation year is t, consider the time-varying load effect distribution S(t) as the maximum value in [0, t], and the time-varying resistance distribution R(t) as the minimum value in [0, t]. At this time, the above formula is transformed into:

[0183]

[0184] By stabilizing the non-stationary random process, the maximum distribution function of the load effect in the t-year operation is obtained as

[0185] F t (x) = [F(x)] n

[0186] Where F(x) is the daily load effect distribution function; n is the number of effective action days in the operation year t, n = d × t, where d is the effective number of days of load action in a year.

[0187] Time-varying reliability analysis of track slab structure under different insulation measures

[0188] Using the key sampling method, the time-varying reliability of the longitudinal crack resistance of the track slab under the insulation measures and the reliability improvement value are shown in Figure 9(a) and Figure 9(b).

[0189] As shown in Figure 9(a), both insulation materials have a significant improvement in reliability. The time-varying reliability of the track slab crack resistance under the rubber-plastic board insulation measure is lower than the target reliability value of 1.5 at 55.5 years, which is 14.8 years longer than the 40.7 years without insulation measures. The polyurethane insulation measure has a significant effect, and the time-varying reliability is 2.57 at the 60th year, which still has a high safety margin. As shown in Figure 9(b), the improvement value of the time-varying reliability of the two insulation measures gradually increases with the increase in service time. The maximum improvement value of the polyurethane insulation measure is 2.18, and the improvement value of the rubber-plastic board insulation measure is 0.86.

[0190] The time-varying reliability and reliability improvement value of the longitudinal crack resistance of the track slab under the insulation measures are shown in Figure 10(a) and Figure 10(b).

[0191] As shown in Figure 10(a), similar to the time-varying reliability law of longitudinal crack resistance, both insulation measures have improved the reliability value to varying degrees. However, since the time-varying reliability value of the transverse crack resistance of the track slab without insulation measures was still not lower than the target reliability value in the 60th year, the insulation measures did not play a key role here. As shown in Figure 10(b), the maximum reliability improvement value of the polyurethane insulation measure is 1.53, and the maximum improvement value of the rubber-plastic board insulation measure is 0.78. At this time, the improvement effect of the polyurethane insulation measure is still better than that of the rubber-plastic board.

[0192] The time-varying reliability and reliability improvement value of the longitudinal bending resistance of the track slab under the insulation measures are shown in Figure 11(a) and Figure 11(b).

[0193] As shown in Figure 11(a), the longitudinal bending reliability of the track slab under the insulation measures of polyurethane and rubber-plastic board has been improved compared with the case without insulation. In the 60th year, it is still 3.82 and 3.70 respectively, which is higher than the target reliability of 3.2, and there is a relatively good safety margin. As shown in Figure 11(b), the reliability improvement value under the insulation measures of polyurethane and rubber-plastic board has the same change trend, which is first increased and then decreased, and the maximum improvement value is 1.87 and 1.68 respectively. Compared with the significant difference in the insulation effect of polyurethane and rubber-plastic board under low temperature conditions, the insulation effect of the two is relatively close under high temperature conditions.

[0194] The time-varying reliability and reliability improvement value of the track slab transverse bending under the insulation measures are shown in Figure 12(a) and Figure 12(b). It can be seen that both the polyurethane and rubber-plastic board insulation measures have a certain improvement on the time-varying reliability of the track slab transverse bending, and the improvement value change curves are relatively close, with the maximum values ​​of 1.36 and 1.29 respectively. Since the transverse bending performance of the track slab has been kept in a good state within the design life, both are higher than the target reliability value, and the insulation measures only increase the safety margin to a certain extent, and the improvement effect is relatively limited at this time.

[0195] This example statistically analyzes the long-term monitoring data of the temperature field of ballastless track in cold regions, obtains the probability distribution of the track structure temperature field under different insulation measures, generates random load samples using the Latin hypercube sampling method, and analyzes the track slab load effect under different insulation measures based on the finite element model. At the same time, combined with the track slab time-varying resistance model, the Monte Carlo key sampling method is used to analyze the time-varying reliability of the track slab anti-cracking and anti-bending under different insulation measures. The main conclusions are as follows:

[0196] (1) Polyurethane foam board and rubber-plastic board insulation materials can reduce the temperature difference of the track structure by about 50% in hot summer weather. In cold winter weather, polyurethane foam board can still reduce the temperature difference by about 40%, while rubber-plastic board can only reduce the temperature difference by 6%.

[0197] (2) Under the coupling effect of train load and positive temperature gradient, the polyurethane foam board and rubber-plastic board insulation measures can reduce the longitudinal bending moment of the structure by 45.8% and 38.7%, respectively, which can effectively weaken the structural load effect under high temperature weather in summer.

[0198] (3) Under the coupling effect of train load and negative temperature gradient, the insulation measures of polyurethane foam board and rubber-plastic board can reduce the longitudinal tensile stress of the structure by 31.1% and 17.0%, respectively. In low temperature weather, the insulation measures still have a good effect and can significantly reduce the tensile stress of the track plate. However, the insulation effect of the rubber-plastic board insulation material is already weak at this time, and the stress reduction of the track plate is less than 20%.

[0199] (4) The longitudinal bending reliability of the track slab under the polyurethane and rubber-plastic board insulation measures increased by 1.56 and 1.37 respectively, and the transverse reliability increased by 0.72 and 0.68 respectively; the longitudinal time-varying reliability of the track slab under the polyurethane and rubber-plastic board measures increased by 3.21 and 1.26 respectively, and the transverse reliability increased by 2.49 and 1.24. The insulation measures can significantly improve the time-varying reliability of the structure, which is more obvious in the longitudinal direction of the structure, and can significantly improve the operational safety of the structure. In addition, in comparison, the improvement effect of polyurethane foam board is better than that of rubber-plastic board in both crack resistance and bending resistance.

[0200] The present invention and its embodiments are described schematically above, and the description is not restrictive. The drawings show only one embodiment of the present invention, and the actual structure is not limited thereto. Therefore, if a person skilled in the art is inspired by it and designs a structural method and an embodiment similar to the technical solution without creativity without departing from the purpose of the invention, they shall all fall within the protection scope of the present invention.

Claims

1. A time-varying reliability analysis method for ballastless track slabs in cold regions under different thermal insulation measures, characterized by: The following steps are involved:

1. Statistical analysis is performed on the long-term monitoring data of the temperature field of ballastless track in cold regions to obtain the probability distribution of the temperature field of the track structure under different insulation measures; In step 1, the test casting model is monitored for a long time. The test casting model is a full-scale model of CRTSⅢ type ballastless track, which includes the bearing groove, C60 prefabricated track plate, C40 self-compacting concrete, and C40 base plate from top to bottom. Seven measuring points are set from the top surface of the track plate to the center of the base plate at the transverse center of the track structure, which are set as Z1 on the top surface of the track plate, Z2 at 1 / 4 of the distance from the track plate, Z3 in the middle of the track plate, Z4 at the junction of the track plate and the self-compacting concrete, Z5 in the middle of the self-compacting concrete layer, Z6 at the bottom of the self-compacting concrete layer, and Z7 in the middle of the base plate. The temperature distribution law inside the track structure is mastered through the seven temperature measuring point positions. Second, the Latin hypercube sampling method is used to generate random load samples, and the load effects of track slabs under different insulation measures are analyzed based on the finite element model; Third, combined with the track slab time-varying resistance model, the Monte Carlo key sampling method was used to analyze the time-varying reliability of the track slab's crack resistance and bending resistance under different insulation measures; The track slab time-varying resistance model includes the track slab anti-cracking time-varying resistance distribution model and the track slab anti-bending time-varying resistance distribution model; The time-varying resistance distribution model of track slab crack resistance includes the concrete strength time-varying model and the prestress time-varying model. In the concrete strength time-varying model, the degradation function of concrete tensile strength over time is expressed as: Where R c0 Represents the standard value of the initial tensile strength of concrete, and the fitting function The expression is: Where: ε is the degradation coefficient, t is the time; In the prestress time-varying model, the prediction formula for the prestress loss of ballastless track slab is: Where: R p0 is the initial value of precompression stress; The time-varying resistance distribution model of track slab bending is: In the formula is the degradation function of the track slab bending strength, expressed as: Where t is the service time of ballastless track, T0 is the starting time of resistance degradation, in years; k1 and k2 are resistance degradation rate parameters; Initial bending bearing capacity of track slab M R0 The expression is: Where: f s 、f s ' and f p are the design values ​​of tensile strength of ordinary steel bars in the tension zone, compressive strength of ordinary steel bars in the compression zone, and tensile strength of prestressed steel bars; A s , A s 'With A p are the cross-sectional areas of ordinary steel bars and prestressed bars in the tension zone and compression zone respectively; f c is the design value of concrete axial compressive strength; b is the width of the bending section; h0 is the effective height of the section; a s ' is the distance from the steel bars in the compression zone to the edge of the section.

2. The method for analyzing the time-varying reliability of ballastless track slabs in cold regions under different thermal insulation measures according to claim 1 is characterized by: In step 1, the temperature difference data of the track slab and the self-compacting concrete layer over multiple days are selected, and the frequency distribution histogram of the maximum positive temperature difference of the track structure, the frequency distribution histogram of the maximum negative temperature difference of the track structure and the corresponding Weibull distribution fitting curve are drawn and analyzed.

3. The method for analyzing the time-varying reliability of ballastless track slabs in cold regions under different thermal insulation measures according to claim 2 is characterized by: In step 2, the process of the Latin hypercube sampling method is as follows: Assume that the number of random variables is K, P is an N×K matrix, each column is a different random integer from 1 to N, the matrix R is an N×K matrix, and the column elements in R are composed of uniform random numbers from 0 to 1, where N is the number of sample groups selected, then the sampling matrix S is: The sampling matrix contains random values ​​from 0 to 1, which are uniformly distributed in the stratified region of the cumulative distribution function; ij is the i-th row and j-th column element of matrix S, then: Where: is the jth variable of the i-th group of samples obtained by Latin hypercube sampling, It is the distribution function that the j-th variable of the sampling matrix S conforms to; Through the above steps, stratified sampling can be performed on multiple groups of random variables to generate random load samples.

4. The method for analyzing the time-varying reliability of ballastless track slabs in cold regions under different thermal insulation measures according to claim 3 is characterized by: In step 2, based on the train load and positive and negative temperature gradient random variable samples, they are input into the finite element model for batch calculation. The maximum value of each load effect of the track plate is extracted under each group of samples, and the frequency distribution histogram is drawn to analyze the track plate load effect.

5. The method for analyzing time-varying reliability of ballastless track slabs in cold regions under different thermal insulation measures according to claim 4 is characterized by: The Monte Carlo key sampling method is: Let X = (X1, X2, ···X1) be a structural random variable, whose joint probability density is f X (x) = f X (x1,x2...x n ), the structural function function Z = g(x), then the failure probability P f Written as the mathematical expectation value of the failure domain characteristic function: Where: R n is an n-dimensional vector space; is an n-dimensional vector of random variable combinations (x1, x2...x n ) T ; is the failure domain indicative function, expressed as: According to the law of large numbers, the sample mean of the failure domain characteristic function approximately represents the expected value of the failure domain characteristic function, that is, P f for: At this time, by constructing an important sampling function, the centroid of the sample distribution is controlled near the failure domain, reducing the number of sampling times in the above formula; for the selection of the sampling centroid, the design verification point of the functional function is currently selected, and the important sampling function is constructed with this as the center. Let the important sampling density function be h Y (·), then the failure probability P f Translates to: After N samples are extracted according to the above formula, N sample vectors are obtained. Then we get P f The approximate value is:

6. The method for analyzing the time-varying reliability of ballastless track slabs in cold regions under different thermal insulation measures according to claim 5 is characterized by: The time-varying reliability analysis method is: The performance function Z is: Z=g(R,S)=RS Where: R is the structural resistance; S is the structural load effect, that is, the deformation and displacement caused by various loads acting on the structure; The structural reliability of ballastless track will change with the service time t within the design life, so the structural performance function Z needs to be considered as a time-dependent random process. In this case, the above formula can be rewritten as: Z(t)=g(R(t),S(t))=R(t)-S(t) Where: t is the time variable; R(t) is the time-varying resistance; S(t) is the time-varying load effect; Assume that the operation year is t, consider the time-varying load effect distribution S(t) as the maximum value in [0, t], and the time-varying resistance distribution R(t) as the minimum value in [0, t]. At this time, the above formula is transformed into: By stabilizing the non-stationary random process, the maximum distribution function of the load effect in the t-year operation is obtained as follows: F t (x)=[F(x)] n Where F(x) is the daily load effect distribution function; n is the number of effective action days in the operation year t, n = d × t, where d is the effective number of days of load action in a year.

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